(* Title: HOL/Library/Float.thy
Author: Johannes Hölzl, Fabian Immler
Copyright 2012 TU München
*)
header {* Floating-Point Numbers *}
theory Float
imports Complex_Main Lattice_Algebras
begin
definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"
typedef float = float
morphisms real_of_float float_of
unfolding float_def by auto
defs (overloaded)
real_of_float_def[code_unfold]: "real \<equiv> real_of_float"
lemma type_definition_float': "type_definition real float_of float"
using type_definition_float unfolding real_of_float_def .
setup_lifting (no_code) type_definition_float'
lemmas float_of_inject[simp]
declare [[coercion "real :: float \<Rightarrow> real"]]
lemma real_of_float_eq:
fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"
unfolding real_of_float_def real_of_float_inject ..
lemma float_of_real[simp]: "float_of (real x) = x"
unfolding real_of_float_def by (rule real_of_float_inverse)
lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
unfolding real_of_float_def by (rule float_of_inverse)
subsection {* Real operations preserving the representation as floating point number *}
lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
by (auto simp: float_def)
lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)
lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp
lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp
lemma neg_numeral_float[simp]: "- numeral i \<in> float" by (intro floatI[of "- numeral i" 0]) simp
lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp
lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp
lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp
lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp
lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp
lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float" by (intro floatI[of 1 "- numeral i"]) simp
lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp
lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"
unfolding float_def
proof (safe, simp)
fix e1 m1 e2 m2 :: int
{ fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"
then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)
then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
by blast }
note * = this
show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
proof (cases e1 e2 rule: linorder_le_cases)
assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
qed (rule *)
qed
lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"
apply (auto simp: float_def)
apply (rule_tac x="-x" in exI)
apply (rule_tac x="xa" in exI)
apply (simp add: field_simps)
done
lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"
apply (auto simp: float_def)
apply (rule_tac x="x * xa" in exI)
apply (rule_tac x="xb + xc" in exI)
apply (simp add: powr_add)
done
lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"
using plus_float [of x "- y"] by simp
lemma abs_float[simp]: "x \<in> float \<Longrightarrow> abs x \<in> float"
by (cases x rule: linorder_cases[of 0]) auto
lemma sgn_of_float[simp]: "x \<in> float \<Longrightarrow> sgn x \<in> float"
by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)
lemma div_power_2_float[simp]: "x \<in> float \<Longrightarrow> x / 2^d \<in> float"
apply (auto simp add: float_def)
apply (rule_tac x="x" in exI)
apply (rule_tac x="xa - d" in exI)
apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
done
lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"
apply (auto simp add: float_def)
apply (rule_tac x="x" in exI)
apply (rule_tac x="xa - d" in exI)
apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])
done
lemma div_numeral_Bit0_float[simp]:
assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"
proof -
have "(x / numeral n) / 2^1 \<in> float"
by (intro x div_power_2_float)
also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"
by (induct n) auto
finally show ?thesis .
qed
lemma div_neg_numeral_Bit0_float[simp]:
assumes x: "x / numeral n \<in> float" shows "x / (- numeral (Num.Bit0 n)) \<in> float"
proof -
have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp
also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)"
by simp
finally show ?thesis .
qed
lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp
declare Float.rep_eq[simp]
lemma compute_real_of_float[code]:
"real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"
by (simp add: real_of_float_def[symmetric] powr_int)
code_datatype Float
subsection {* Arithmetic operations on floating point numbers *}
instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}"
begin
lift_definition zero_float :: float is 0 by simp
declare zero_float.rep_eq[simp]
lift_definition one_float :: float is 1 by simp
declare one_float.rep_eq[simp]
lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp
declare plus_float.rep_eq[simp]
lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp
declare times_float.rep_eq[simp]
lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op -" by simp
declare minus_float.rep_eq[simp]
lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp
declare uminus_float.rep_eq[simp]
lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp
declare abs_float.rep_eq[simp]
lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp
declare sgn_float.rep_eq[simp]
lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" .
lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" .
declare less_eq_float.rep_eq[simp]
lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" .
declare less_float.rep_eq[simp]
instance
proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+
end
lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n"
by (induct n) simp_all
lemma fixes x y::float
shows real_of_float_min: "real (min x y) = min (real x) (real y)"
and real_of_float_max: "real (max x y) = max (real x) (real y)"
by (simp_all add: min_def max_def)
instance float :: unbounded_dense_linorder
proof
fix a b :: float
show "\<exists>c. a < c"
apply (intro exI[of _ "a + 1"])
apply transfer
apply simp
done
show "\<exists>c. c < a"
apply (intro exI[of _ "a - 1"])
apply transfer
apply simp
done
assume "a < b"
then show "\<exists>c. a < c \<and> c < b"
apply (intro exI[of _ "(a + b) * Float 1 -1"])
apply transfer
apply (simp add: powr_minus)
done
qed
instantiation float :: lattice_ab_group_add
begin
definition inf_float::"float\<Rightarrow>float\<Rightarrow>float"
where "inf_float a b = min a b"
definition sup_float::"float\<Rightarrow>float\<Rightarrow>float"
where "sup_float a b = max a b"
instance
by default
(transfer, simp_all add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+
end
lemma float_numeral[simp]: "real (numeral x :: float) = numeral x"
apply (induct x)
apply simp
apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float
plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float)
done
lemma transfer_numeral [transfer_rule]:
"rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"
unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp
lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x"
by simp
lemma transfer_neg_numeral [transfer_rule]:
"rel_fun (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"
unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp
lemma
shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"
and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
unfolding real_of_float_eq by simp_all
subsection {* Represent floats as unique mantissa and exponent *}
lemma int_induct_abs[case_names less]:
fixes j :: int
assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"
shows "P j"
proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)
case less show ?case by (rule H[OF less]) simp
qed
lemma int_cancel_factors:
fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"
proof (induct n rule: int_induct_abs)
case (less n)
{ fix m assume n: "n \<noteq> 0" "n = m * r"
then have "\<bar>m \<bar> < \<bar>n\<bar>"
by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)
dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le
mult_eq_0_iff zdvd_mult_cancel1)
from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
then show ?case
by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)
qed
lemma mult_powr_eq_mult_powr_iff_asym:
fixes m1 m2 e1 e2 :: int
assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"
shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
proof
have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
by (simp add: powr_divide2[symmetric] field_simps)
also have "\<dots> = m2 * 2^nat (e2 - e1)"
by (simp add: powr_realpow)
finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
unfolding real_of_int_inject .
with m1 have "m1 = m2"
by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
then show "m1 = m2 \<and> e1 = e2"
using eq `m1 \<noteq> 0` by (simp add: powr_inj)
qed simp
lemma mult_powr_eq_mult_powr_iff:
fixes m1 m2 e1 e2 :: int
shows "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"
using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]
using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]
by (cases e1 e2 rule: linorder_le_cases) auto
lemma floatE_normed:
assumes x: "x \<in> float"
obtains (zero) "x = 0"
| (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"
proof atomize_elim
{ assume "x \<noteq> 0"
from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
by auto
with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
(simp add: powr_add powr_realpow) }
then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
by blast
qed
lemma float_normed_cases:
fixes f :: float
obtains (zero) "f = 0"
| (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"
proof (atomize_elim, induct f)
case (float_of y) then show ?case
by (cases rule: floatE_normed) (auto simp: zero_float_def)
qed
definition mantissa :: "float \<Rightarrow> int" where
"mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
definition exponent :: "float \<Rightarrow> int" where
"exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"
lemma
shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)
and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)
proof -
have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto
then show ?E ?M
by (auto simp add: mantissa_def exponent_def zero_float_def)
qed
lemma
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")
proof cases
assume [simp]: "f \<noteq> (float_of 0)"
have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"
proof (cases f rule: float_normed_cases)
case (powr m e)
then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)
\<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"
by auto
then show ?thesis
unfolding exponent_def mantissa_def
by (rule someI2_ex) (simp add: zero_float_def)
qed (simp add: zero_float_def)
then show ?E ?D by auto
qed simp
lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"
using mantissa_not_dvd[of f] by auto
lemma
fixes m e :: int
defines "f \<equiv> float_of (m * 2 powr e)"
assumes dvd: "\<not> 2 dvd m"
shows mantissa_float: "mantissa f = m" (is "?M")
and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")
proof cases
assume "m = 0" with dvd show "mantissa f = m" by auto
next
assume "m \<noteq> 0"
then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)
from mantissa_exponent[of f]
have "m * 2 powr e = mantissa f * 2 powr exponent f"
by (auto simp add: f_def)
then show "?M" "?E"
using mantissa_not_dvd[OF f_not_0] dvd
by (auto simp: mult_powr_eq_mult_powr_iff)
qed
subsection {* Compute arithmetic operations *}
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
unfolding real_of_float_eq mantissa_exponent[of f] by simp
lemma Float_cases[case_names Float, cases type: float]:
fixes f :: float
obtains (Float) m e :: int where "f = Float m e"
using Float_mantissa_exponent[symmetric]
by (atomize_elim) auto
lemma denormalize_shift:
assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"
obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"
proof
from mantissa_exponent[of f] f_def
have "m * 2 powr e = mantissa f * 2 powr exponent f"
by simp
then have eq: "m = mantissa f * 2 powr (exponent f - e)"
by (simp add: powr_divide2[symmetric] field_simps)
moreover
have "e \<le> exponent f"
proof (rule ccontr)
assume "\<not> e \<le> exponent f"
then have pos: "exponent f < e" by simp
then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"
by simp
also have "\<dots> = 1 / 2^nat (e - exponent f)"
using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])
finally have "m * 2^nat (e - exponent f) = real (mantissa f)"
using eq by simp
then have "mantissa f = m * 2^nat (e - exponent f)"
unfolding real_of_int_inject by simp
with `exponent f < e` have "2 dvd mantissa f"
apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
apply (cases "nat (e - exponent f)")
apply auto
done
then show False using mantissa_not_dvd[OF not_0] by simp
qed
ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
by (simp add: powr_realpow[symmetric])
with `e \<le> exponent f`
show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
unfolding real_of_int_inject by auto
qed
lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
by transfer simp
hide_fact (open) compute_float_zero
lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
by transfer simp
hide_fact (open) compute_float_one
definition normfloat :: "float \<Rightarrow> float" where
[simp]: "normfloat x = x"
lemma compute_normfloat[code]: "normfloat (Float m e) =
(if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))
else if m = 0 then 0 else Float m e)"
unfolding normfloat_def
by transfer (auto simp add: powr_add zmod_eq_0_iff)
hide_fact (open) compute_normfloat
lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
by transfer simp
hide_fact (open) compute_float_numeral
lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
by transfer simp
hide_fact (open) compute_float_neg_numeral
lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
by transfer simp
hide_fact (open) compute_float_uminus
lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"
by transfer (simp add: field_simps powr_add)
hide_fact (open) compute_float_times
lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =
(if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else
if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1
else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"
by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])
hide_fact (open) compute_float_plus
lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"
by simp
hide_fact (open) compute_float_minus
lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"
by transfer (simp add: sgn_times)
hide_fact (open) compute_float_sgn
lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .
lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])
hide_fact (open) compute_is_float_pos
lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"
by transfer (simp add: field_simps)
hide_fact (open) compute_float_less
lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .
lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])
hide_fact (open) compute_is_float_nonneg
lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"
by transfer (simp add: field_simps)
hide_fact (open) compute_float_le
lift_definition is_float_zero :: "float \<Rightarrow> bool" is "op = 0 :: real \<Rightarrow> bool" .
lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"
by transfer (auto simp add: is_float_zero_def)
hide_fact (open) compute_is_float_zero
lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"
by transfer (simp add: abs_mult)
hide_fact (open) compute_float_abs
lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"
by transfer simp
hide_fact (open) compute_float_eq
subsection {* Rounding Real numbers *}
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
"round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where
"round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"
lemma round_down_float[simp]: "round_down prec x \<in> float"
unfolding round_down_def
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
lemma round_up_float[simp]: "round_up prec x \<in> float"
unfolding round_up_def
by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)
lemma round_up: "x \<le> round_up prec x"
by (simp add: powr_minus_divide le_divide_eq round_up_def)
lemma round_down: "round_down prec x \<le> x"
by (simp add: powr_minus_divide divide_le_eq round_down_def)
lemma round_up_0[simp]: "round_up p 0 = 0"
unfolding round_up_def by simp
lemma round_down_0[simp]: "round_down p 0 = 0"
unfolding round_down_def by simp
lemma round_up_diff_round_down:
"round_up prec x - round_down prec x \<le> 2 powr -prec"
proof -
have "round_up prec x - round_down prec x =
(ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"
by (simp add: round_up_def round_down_def field_simps)
also have "\<dots> \<le> 1 * 2 powr -prec"
by (rule mult_mono)
(auto simp del: real_of_int_diff
simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)
finally show ?thesis by simp
qed
lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"
unfolding round_down_def
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
(simp add: powr_add[symmetric])
lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"
unfolding round_up_def
by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])
(simp add: powr_add[symmetric])
subsection {* Rounding Floats *}
lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp
declare float_up.rep_eq[simp]
lemma round_up_correct:
shows "round_up e f - f \<in> {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp
also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
finally show "round_up e f - f \<le> 2 powr real (- e)"
by simp
qed (simp add: algebra_simps round_up)
lemma float_up_correct:
shows "real (float_up e f) - real f \<in> {0..2 powr -e}"
by transfer (rule round_up_correct)
lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp
declare float_down.rep_eq[simp]
lemma round_down_correct:
shows "f - (round_down e f) \<in> {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp
also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp
finally show "f - round_down e f \<le> 2 powr real (- e)"
by simp
qed (simp add: algebra_simps round_down)
lemma float_down_correct:
shows "real f - real (float_down e f) \<in> {0..2 powr -e}"
by transfer (rule round_down_correct)
lemma compute_float_down[code]:
"float_down p (Float m e) =
(if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"
proof cases
assume "p + e < 0"
hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
using powr_realpow[of 2 "nat (-(p + e))"] by simp
also have "... = 1 / 2 powr p / 2 powr e"
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
finally show ?thesis
using `p + e < 0`
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
next
assume "\<not> p + e < 0"
then have r: "real e + real p = real (nat (e + p))" by simp
have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
by (auto intro: exI[where x="m*2^nat (e+p)"]
simp add: ac_simps powr_add[symmetric] r powr_realpow)
with `\<not> p + e < 0` show ?thesis
by transfer
(auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)
qed
hide_fact (open) compute_float_down
lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"
using round_down_correct[of f e] by simp
lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"
using round_up_correct[of e f] by simp
lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"
by (auto simp: round_down_def)
lemma ceil_divide_floor_conv:
assumes "b \<noteq> 0"
shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"
proof cases
assume "\<not> b dvd a"
hence "a mod b \<noteq> 0" by auto
hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
proof -
have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
qed
thus ?thesis using `\<not> b dvd a` by simp
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
lemma compute_float_up[code]:
"float_up p (Float m e) =
(let P = 2^nat (-(p + e)); r = m mod P in
if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"
proof cases
assume "p + e < 0"
hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"
using powr_realpow[of 2 "nat (-(p + e))"] by simp
also have "... = 1 / 2 powr p / 2 powr e"
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
finally have twopow_rewrite:
"real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .
with `p + e < 0` have powr_rewrite:
"2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"
unfolding powr_divide2 by simp
show ?thesis
proof cases
assume "2^nat (-(p + e)) dvd m"
with `p + e < 0` twopow_rewrite show ?thesis
by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)
next
assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"
have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =
real m / real ((2::int) ^ nat (- (p + e)))"
by (simp add: field_simps)
have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =
real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"
using ndvd unfolding powr_rewrite one_div
by (subst ceil_divide_floor_conv) (auto simp: field_simps)
thus ?thesis using `p + e < 0` twopow_rewrite
by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])
qed
next
assume "\<not> p + e < 0"
then have r1: "real e + real p = real (nat (e + p))" by simp
have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"
by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow
intro: exI[where x="m*2^nat (e+p)"])
then show ?thesis using `\<not> p + e < 0`
by transfer
(simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)
qed
hide_fact (open) compute_float_up
lemmas real_of_ints =
real_of_int_zero
real_of_one
real_of_int_add
real_of_int_minus
real_of_int_diff
real_of_int_mult
real_of_int_power
real_numeral
lemmas real_of_nats =
real_of_nat_zero
real_of_nat_one
real_of_nat_1
real_of_nat_add
real_of_nat_mult
real_of_nat_power
lemmas int_of_reals = real_of_ints[symmetric]
lemmas nat_of_reals = real_of_nats[symmetric]
lemma two_real_int: "(2::real) = real (2::int)" by simp
lemma two_real_nat: "(2::real) = real (2::nat)" by simp
lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp
subsection {* Compute bitlen of integers *}
definition bitlen :: "int \<Rightarrow> int" where
"bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
lemma bitlen_nonneg: "0 \<le> bitlen x"
proof -
{
assume "0 > x"
have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
also have "... < log 2 (-x)" using `0 > x` by auto
finally have "-1 < log 2 (-x)" .
} thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
qed
lemma bitlen_bounds:
assumes "x > 0"
shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
proof
have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
using real_nat_eq_real[of "floor (log 2 (real x))"]
by simp
also have "... \<le> 2 powr log 2 (real x)"
by simp
also have "... = real x"
using `0 < x` by simp
finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
by (simp add: bitlen_def)
next
have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
apply (simp add: powr_realpow[symmetric])
using `x > 0` by simp
finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)
qed
lemma bitlen_pow2[simp]:
assumes "b > 0"
shows "bitlen (b * 2 ^ c) = bitlen b + c"
proof -
from assms have "b * 2 ^ c > 0" by (auto intro: mult_pos_pos)
thus ?thesis
using floor_add[of "log 2 b" c] assms
by (auto simp add: log_mult log_nat_power bitlen_def)
qed
lemma bitlen_Float:
fixes m e
defines "f \<equiv> Float m e"
shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
proof (cases "m = 0")
case True
then show ?thesis by (simp add: f_def bitlen_def Float_def)
next
case False
hence "f \<noteq> float_of 0"
unfolding real_of_float_eq by (simp add: f_def)
hence "mantissa f \<noteq> 0"
by (simp add: mantissa_noteq_0)
moreover
obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`])
ultimately show ?thesis by (simp add: abs_mult)
qed
lemma compute_bitlen[code]:
shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
proof -
{ assume "2 \<le> x"
then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"
by (simp add: log_mult zmod_zdiv_equality')
also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"
proof cases
assume "x mod 2 = 0" then show ?thesis by simp
next
def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
then have "0 \<le> n"
using `2 \<le> x` by simp
assume "x mod 2 \<noteq> 0"
with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
moreover
{ have "real (2^nat n :: int) = 2 powr (nat n)"
by (simp add: powr_realpow)
also have "\<dots> \<le> 2 powr (log 2 x)"
using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
ultimately have "2^nat n \<le> x - 1" by simp
then have "2^nat n \<le> real (x - 1)"
unfolding real_of_int_le_iff[symmetric] by simp
{ have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
using `0 \<le> n` by (simp add: log_nat_power)
also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
unfolding n_def `x mod 2 = 1` by auto
qed
finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
moreover
{ assume "x < 2" "0 < x"
then have "x = 1" by simp
then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }
ultimately show ?thesis
unfolding bitlen_def
by (auto simp: pos_imp_zdiv_pos_iff not_le)
qed
hide_fact (open) compute_bitlen
lemma float_gt1_scale: assumes "1 \<le> Float m e"
shows "0 \<le> e + (bitlen m - 1)"
proof -
have "0 < Float m e" using assms by auto
hence "0 < m" using powr_gt_zero[of 2 e]
by (auto simp: zero_less_mult_iff)
hence "m \<noteq> 0" by auto
show ?thesis
proof (cases "0 \<le> e")
case True thus ?thesis using `0 < m` by (simp add: bitlen_def)
next
have "(1::int) < 2" by simp
case False let ?S = "2^(nat (-e))"
have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus field_simps inverse_eq_divide)
hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus)
hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hence "?S \<le> real m" unfolding mult_assoc by auto
hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
hence "-e < bitlen m" using False by auto
thus ?thesis by auto
qed
qed
lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
proof -
let ?B = "2^nat(bitlen m - 1)"
have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
thus "1 \<le> real m / ?B" by auto
have "m \<noteq> 0" using assms by auto
have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
thus "real m / ?B < 2" by auto
qed
subsection {* Approximation of positive rationals *}
lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"
by (simp add: zdiv_zmult2_eq)
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
lemma real_div_nat_eq_floor_of_divide:
fixes a b::nat
shows "a div b = real (floor (a/b))"
by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)
definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"
lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp
lemma compute_lapprox_posrat[code]:
fixes prec x y
shows "lapprox_posrat prec x y =
(let
l = rat_precision prec x y;
d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y
in normfloat (Float d (- l)))"
unfolding div_mult_twopow_eq normfloat_def
by transfer
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
del: two_powr_minus_int_float)
hide_fact (open) compute_lapprox_posrat
lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"
is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp
(* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)
lemma compute_rapprox_posrat[code]:
fixes prec x y
defines "l \<equiv> rat_precision prec x y"
shows "rapprox_posrat prec x y = (let
l = l ;
X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;
d = fst X div snd X ;
m = fst X mod snd X
in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"
proof (cases "y = 0")
assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp
next
assume "y \<noteq> 0"
show ?thesis
proof (cases "0 \<le> l")
assume "0 \<le> l"
def x' == "x * 2 ^ nat l"
have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
moreover have "real x * 2 powr real l = real x'"
by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
ultimately show ?thesis
unfolding normfloat_def
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
l_def[symmetric, THEN meta_eq_to_obj_eq]
by transfer
(simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
next
assume "\<not> 0 \<le> l"
def y' == "y * 2 ^ nat (- l)"
from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
moreover have "real x * real (2::int) powr real l / real y = x / real y'"
using `\<not> 0 \<le> l`
by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)
ultimately show ?thesis
unfolding normfloat_def
using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
l_def[symmetric, THEN meta_eq_to_obj_eq]
by transfer
(simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)
qed
qed
hide_fact (open) compute_rapprox_posrat
lemma rat_precision_pos:
assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
shows "rat_precision n (int x) (int y) > 0"
proof -
{ assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }
hence "bitlen (int x) < bitlen (int y)" using assms
by (simp add: bitlen_def del: floor_add_one)
(auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)
thus ?thesis
using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
qed
lemma power_aux: assumes "x > 0" shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"
proof -
def y \<equiv> "nat (x - 1)" moreover
have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp
ultimately show ?thesis using assms by simp
qed
lemma rapprox_posrat_less1:
assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
shows "real (rapprox_posrat n x y) < 1"
proof -
have powr1: "2 powr real (rat_precision n (int x) (int y)) =
2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms
by (simp add: powr_realpow[symmetric])
have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *
2 powr real (rat_precision n (int x) (int y))" by simp
also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"
apply (rule mult_strict_right_mono) by (insert assms) auto
also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"
using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_minus)
also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"
using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])
also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"
unfolding int_of_reals real_of_int_le_iff
using rat_precision_pos[OF assms] by (rule power_aux)
finally show ?thesis
apply (transfer fixing: n x y)
apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1)
unfolding int_of_reals real_of_int_less_iff
apply (simp add: ceiling_less_eq)
done
qed
lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
"\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
lemma compute_lapprox_rat[code]:
"lapprox_rat prec x y =
(if y = 0 then 0
else if 0 \<le> x then
(if 0 < y then lapprox_posrat prec (nat x) (nat y)
else - (rapprox_posrat prec (nat x) (nat (-y))))
else (if 0 < y
then - (rapprox_posrat prec (nat (-x)) (nat y))
else lapprox_posrat prec (nat (-x)) (nat (-y))))"
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hide_fact (open) compute_lapprox_rat
lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is
"\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp
lemma compute_rapprox_rat[code]:
"rapprox_rat prec x y =
(if y = 0 then 0
else if 0 \<le> x then
(if 0 < y then rapprox_posrat prec (nat x) (nat y)
else - (lapprox_posrat prec (nat x) (nat (-y))))
else (if 0 < y
then - (lapprox_posrat prec (nat (-x)) (nat y))
else rapprox_posrat prec (nat (-x)) (nat (-y))))"
by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hide_fact (open) compute_rapprox_rat
subsection {* Division *}
definition "real_divl prec a b = round_down (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"
definition "real_divr prec a b = round_up (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"
lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl
by (simp add: real_divl_def)
lemma compute_float_divl[code]:
"float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
proof cases
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
by (simp add: field_simps powr_divide2[symmetric])
show ?thesis
using not_0
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,
simp add: field_simps)
qed (transfer, auto simp: real_divl_def)
hide_fact (open) compute_float_divl
lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr
by (simp add: real_divr_def)
lemma compute_float_divr[code]:
"float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
proof cases
let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"
let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"
assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"
then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"
by (simp add: abs_mult log_mult rat_precision_def bitlen_def)
have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"
by (simp add: field_simps powr_divide2[symmetric])
show ?thesis
using not_0
by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def,
simp add: field_simps)
qed (transfer, auto simp: real_divr_def)
hide_fact (open) compute_float_divr
subsection {* Lemmas needed by Approximate *}
lemma Float_num[simp]: shows
"real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
"real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
"real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"] two_powr_int_float[of "-3"]
using powr_realpow[of 2 2] powr_realpow[of 2 3]
using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]
by auto
lemma real_of_Float_int[simp]: "real (Float n 0) = real n" by simp
lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
by arith
lemma lapprox_rat:
shows "real (lapprox_rat prec x y) \<le> real x / real y"
using round_down by (simp add: lapprox_rat_def)
lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"
proof -
from zmod_zdiv_equality'[of a b]
have "a = b * (a div b) + a mod b" by simp
also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)
using assms by simp
finally show ?thesis by simp
qed
lemma lapprox_rat_nonneg:
fixes n x y
defines "p == int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"
assumes "0 \<le> x" "0 < y"
shows "0 \<le> real (lapprox_rat n x y)"
using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]
powr_int[of 2, simplified]
by (auto simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)
lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
using round_up by (simp add: rapprox_rat_def)
lemma rapprox_rat_le1:
fixes n x y
assumes xy: "0 \<le> x" "0 < y" "x \<le> y"
shows "real (rapprox_rat n x y) \<le> 1"
proof -
have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"
using xy unfolding bitlen_def by (auto intro!: floor_mono)
then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)
have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>
\<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"
using xy by (auto intro!: ceiling_mono simp: field_simps)
also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"
using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`
by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)
finally show ?thesis
by (simp add: rapprox_rat_def round_up_def)
(simp add: powr_minus inverse_eq_divide)
qed
lemma rapprox_rat_nonneg_neg:
"0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
unfolding rapprox_rat_def round_up_def
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
lemma rapprox_rat_neg:
"x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
unfolding rapprox_rat_def round_up_def
by (auto simp: field_simps mult_le_0_iff)
lemma rapprox_rat_nonpos_pos:
"x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real (rapprox_rat n x y) \<le> 0"
unfolding rapprox_rat_def round_up_def
by (auto simp: field_simps mult_le_0_iff)
lemma real_divl: "real_divl prec x y \<le> x / y"
by (simp add: real_divl_def round_down)
lemma real_divr: "x / y \<le> real_divr prec x y"
using round_up by (simp add: real_divr_def)
lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
by transfer (rule real_divl)
lemma real_divl_lower_bound:
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"
by (simp add: real_divl_def round_down_def zero_le_mult_iff zero_le_divide_iff)
lemma float_divl_lower_bound:
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"
by transfer (rule real_divl_lower_bound)
lemma exponent_1: "exponent 1 = 0"
using exponent_float[of 1 0] by (simp add: one_float_def)
lemma mantissa_1: "mantissa 1 = 1"
using mantissa_float[of 1 0] by (simp add: one_float_def)
lemma bitlen_1: "bitlen 1 = 1"
by (simp add: bitlen_def)
lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"
proof
assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp
show "x = 0" by (simp add: zero_float_def z)
qed (simp add: zero_float_def)
lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"
proof (cases "x = 0", simp)
assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto
have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]
real_of_int_le_iff less_imp_le)
finally show ?thesis by (simp add: powr_add)
qed
lemma real_divl_pos_less1_bound:
"0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x"
proof (unfold real_divl_def)
fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec"
def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>"
show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) "
proof cases
assume nonneg: "0 \<le> p"
hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"
by (simp add: powr_int del: real_of_int_power) simp
also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp
also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>
floor (real ((2::int) ^ nat p) * (1 / x))"
by (rule le_mult_floor) (auto simp: x prec less_imp_le)
finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)
thus ?thesis unfolding p_def[symmetric]
using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)
next
assume neg: "\<not> 0 \<le> p"
have "x = 2 powr (log 2 x)"
using x by simp
also have "2 powr (log 2 x) \<le> 2 powr p"
proof (rule powr_mono)
have "log 2 x \<le> \<lceil>log 2 x\<rceil>"
by simp
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"
using ceiling_diff_floor_le_1[of "log 2 x"] by simp
also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"
using prec by simp
finally show "log 2 x \<le> real p"
using x by (simp add: p_def)
qed simp
finally have x_le: "x \<le> 2 powr p" .
from neg have "2 powr real p \<le> 2 powr 0"
by (intro powr_mono) auto
also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp
also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff
using x x_le by (intro floor_mono) (simp add: pos_le_divide_eq mult_pos_pos)
finally show ?thesis
using prec x unfolding p_def[symmetric]
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos)
qed
qed
lemma float_divl_pos_less1_bound:
"0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"
by (transfer, rule real_divl_pos_less1_bound)
lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
by transfer (rule real_divr)
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> real_divr prec 1 x"
proof -
have "1 \<le> 1 / x" using `0 < x` and `x < 1` by auto
also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
finally show ?thesis by auto
qed
lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"
by transfer (rule real_divr_pos_less1_lower_bound)
lemma real_divr_nonpos_pos_upper_bound:
"x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real_divr prec x y \<le> 0"
by (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def real_divr_def)
lemma float_divr_nonpos_pos_upper_bound:
"real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"
by transfer (rule real_divr_nonpos_pos_upper_bound)
lemma real_divr_nonneg_neg_upper_bound:
"0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real_divr prec x y \<le> 0"
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def real_divr_def)
lemma float_divr_nonneg_neg_upper_bound:
"0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"
by transfer (rule real_divr_nonneg_neg_upper_bound)
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
"truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
lemma truncate_down: "truncate_down prec x \<le> x"
using round_down by (simp add: truncate_down_def)
lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"
by (rule order_trans[OF truncate_down])
definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where
"truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
lemma truncate_up: "x \<le> truncate_up prec x"
using round_up by (simp add: truncate_up_def)
lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"
by (rule order_trans[OF _ truncate_up])
lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"
by (simp add: truncate_up_def)
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
by (simp add: truncate_up_def)
lemma float_round_up: "real x \<le> real (float_round_up prec x)"
using truncate_up by transfer simp
lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down
by (simp add: truncate_down_def)
lemma float_round_down: "real (float_round_down prec x) \<le> real x"
using truncate_down by transfer simp
lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"
using floor_add[of x i] by (simp del: floor_add add: ac_simps)
lemma compute_float_round_down[code]:
"float_round_down prec (Float m e) = (let d = bitlen (abs m) - int prec in
if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
else Float m e)"
using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def
cong del: if_weak_cong)
hide_fact (open) compute_float_round_down
lemma compute_float_round_up[code]:
"float_round_up prec (Float m e) = (let d = (bitlen (abs m) - int prec) in
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P
in Float (n + (if r = 0 then 0 else 1)) (e + d)
else Float m e)"
using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]
unfolding Let_def
by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_up_def
cong del: if_weak_cong)
hide_fact (open) compute_float_round_up
lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"
by (auto intro!: ceiling_mono simp: round_up_def)
lemma truncate_up_nonneg_mono:
assumes "0 \<le> x" "x \<le> y"
shows "truncate_up prec x \<le> truncate_up prec y"
proof -
{
assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"
hence ?thesis
using assms
by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
} moreover {
assume "0 < x"
hence "log 2 x \<le> log 2 y" using assms by auto
moreover
assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"
ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
unfolding atomize_conj
by (metis floor_less_cancel linorder_cases not_le)
have "truncate_up prec x =
real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1)"
using assms by (simp add: truncate_up_def round_up_def)
also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"
proof (unfold ceiling_le_eq)
have "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))"
using real_of_int_floor_add_one_ge[of "log 2 x"] assms
by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
using `0 < x` by (simp add: powr_realpow)
qed
hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
by (auto simp: powr_realpow)
also
have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
using logless flogless by (auto intro!: floor_mono)
also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
using assms `0 < x`
by (auto simp: algebra_simps)
finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
by simp
also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))"
by (subst powr_add[symmetric]) simp
also have "\<dots> = y"
using `0 < x` assms
by (simp add: powr_add)
also have "\<dots> \<le> truncate_up prec y"
by (rule truncate_up)
finally have ?thesis .
} moreover {
assume "~ 0 < x"
hence ?thesis
using assms
by (auto intro!: truncate_up_le)
} ultimately show ?thesis
by blast
qed
lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"
by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)
lemma truncate_down_nonpos: "x \<le> 0 \<Longrightarrow> truncate_down prec x \<le> 0"
by (auto simp: truncate_down_def round_down_def intro!: mult_nonpos_nonneg
order_le_less_trans[of _ 0, OF mult_nonpos_nonneg])
lemma truncate_up_switch_sign_mono:
assumes "x \<le> 0" "0 \<le> y"
shows "truncate_up prec x \<le> truncate_up prec y"
proof -
note truncate_up_nonpos[OF `x \<le> 0`]
also note truncate_up_le[OF `0 \<le> y`]
finally show ?thesis .
qed
lemma truncate_down_zeroprec_mono:
assumes "0 < x" "x \<le> y"
shows "truncate_down 0 x \<le> truncate_down 0 y"
proof -
have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
using `0 < x`
by (auto simp: inverse_eq_divide field_simps powr_add powr_divide2[symmetric])
also have "\<dots> < 2 powr 0"
using real_of_int_floor_add_one_gt
unfolding neg_less_iff_less
by (intro powr_less_mono) (auto simp: algebra_simps)
finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"
unfolding less_ceiling_eq real_of_int_minus real_of_one
by simp
moreover
have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
using `x > 0` by auto
ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
by simp
also have "\<dots> \<subseteq> {0}" by auto
finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp
with assms show ?thesis
by (auto simp: truncate_down_def round_down_def)
qed
lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"
by (auto simp: truncate_down_def round_down_def)
lemma truncate_down_zero: "truncate_down prec 0 = 0"
by (auto simp: truncate_down_def round_down_def)
lemma truncate_down_switch_sign_mono:
assumes "x \<le> 0" "0 \<le> y"
assumes "x \<le> y"
shows "truncate_down prec x \<le> truncate_down prec y"
proof -
note truncate_down_nonpos[OF `x \<le> 0`]
also note truncate_down_nonneg[OF `0 \<le> y`]
finally show ?thesis .
qed
lemma truncate_up_uminus_truncate_down:
"truncate_up prec x = - truncate_down prec (- x)"
"truncate_up prec (-x) = - truncate_down prec x"
by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
lemma truncate_down_uminus_truncate_up:
"truncate_down prec x = - truncate_up prec (- x)"
"truncate_down prec (-x) = - truncate_up prec x"
by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)
lemma truncate_down_nonneg_mono:
assumes "0 \<le> x" "x \<le> y"
shows "truncate_down prec x \<le> truncate_down prec y"
proof -
{
assume "0 < x" "prec = 0"
with assms have ?thesis
by (simp add: truncate_down_zeroprec_mono)
} moreover {
assume "~ 0 < x"
with assms have "x = 0" "0 \<le> y" by simp_all
hence ?thesis
by (auto simp add: truncate_down_zero intro!: truncate_down_nonneg)
} moreover {
assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
hence ?thesis
using assms
by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
} moreover {
assume "0 < x"
hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto
moreover
assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`]
by (metis floor_less_cancel linorder_cases not_le)
assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
using `0 < y`
by simp
also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
using `0 \<le> y` `0 \<le> x` assms(2)
by (auto intro!: powr_mono mult_pos_pos divide_left_mono
simp: real_of_nat_diff powr_add
powr_divide2[symmetric])
also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
by (auto simp: powr_add)
finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
using `0 \<le> y`
by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
by (auto simp: truncate_down_def round_down_def)
moreover
{
have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp
also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
also
have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
using logless flogless `x > 0` `y > 0`
by (auto intro!: floor_mono)
finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)
} ultimately have ?thesis
by (metis dual_order.trans truncate_down)
} ultimately show ?thesis by blast
qed
lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"
apply (cases "0 \<le> x")
apply (rule truncate_down_nonneg_mono, assumption+)
apply (simp add: truncate_down_uminus_truncate_up)
apply (cases "0 \<le> y")
apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)
done
lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"
by (simp add: truncate_up_uminus_truncate_down truncate_down_mono)
lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"
apply (auto simp: zero_float_def mult_le_0_iff)
using powr_gt_zero[of 2 b] by simp
lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"
unfolding pprt_def sup_float_def max_def sup_real_def by auto
lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"
unfolding nprt_def inf_float_def min_def inf_real_def by auto
lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .
lemma compute_int_floor_fl[code]:
"int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hide_fact (open) compute_int_floor_fl
lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp
lemma compute_floor_fl[code]:
"floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"
by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)
hide_fact (open) compute_floor_fl
lemma floor_fl: "real (floor_fl x) \<le> real x" by transfer simp
lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp
lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"
proof (cases "floor_fl x = float_of 0")
case True
then show ?thesis by (simp add: floor_fl_def)
next
case False
have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp
obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"
by (rule denormalize_shift[OF eq[THEN eq_reflection] False])
then show ?thesis by simp
qed
end